1.5 Limits

# 1.5 Limits - m n ax bx if m< n the horizontal asymptote...

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1.5 Limits The limits of a function describes the behavior of a function near a particular point, not necessarily at that point. If f(x) gets closer and closer to a certain number L as x gets closer and closer to c from both sides of c , then L is the limit of f(x) as x approaches c . It is expressed as lim ( ) x c f x L = . For polynomial functions, ( 29 lim ( ) x c p x p c = . For rational functions, ( 29 ( 29 ( 29 ( 29 ( ) lim , 0 x c p c p x q c q x q c = . If p(c) and q(c) both equal 0, evaluate the rational function for values of x as x approaches c . Reciprocal Power Rules: For constants A and k, with k > 0, lim 0 k x A x →+∞ = and lim 0 k x A x →-∞ = . Limits at Infinity: For even polynomial functions, lim ( ) x p x →±∞ = and lim ( ) x p x →±∞ - = -∞ . For odd polynomial functions: lim ( ) x p x →∞ = ∞ and lim ( ) x p x →-∞ = -∞ ; this is reversed for –p(x). For rational functions, the limits at infinity are equal to the limits for the horizontal or oblique asymptote. Given a rational function

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Unformatted text preview: m n ax bx : if m < n, the horizontal asymptote is a y = 0; if m = n , the horizontal asymptote is a y = a / b ; if m > n , the oblique asymptote is y = a / b x m-n . Use the graphs to find the limits: (1) 2 lim ( ) x f x → =-3 (2) 2 lim ( ) x f x → = No limit 3. 2 lim ( ) x f x → = lim ( ) x f x →±∞ = 3 Find the indicated limits, if it exists: 4. 3 2 3 lim 3 x x x → + =-x 2.9 2.99 2.999 3.000 3.001 3.01 3.1 f(x) (5) 2 2 6 lim 2 x x x x → +-=-x 1.900 1.990 1.999 2.000 2.001 2.010 2.100 f(x) (6) 9 3 lim 9 x x x →-=-x 8.900 8.990 8.999 9.000 9.001 9.010 9.100 f(x) Find ( 29 lim x f x →∞ and ( 29 lim x f x →-∞ : (7) f(x) = 1 – x + 2x 2 – 3x 3 x 1,000 10,000 100,000 1,000,000 f(x) x-1,000-10,000-100,000-1,000,000 f(x) lim ( ) x f x →∞ = lim ( ) x f x →-∞ = (8) ( 29 3 3 1 3 2 6 2 x f x x x-=-+ Homework: 1-48, 54, 55, Pages 69-72...
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1.5 Limits - m n ax bx if m< n the horizontal asymptote...

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